Related papers: Polarized Hodge Structures for Clemens Manifolds

We determine several classes of smooth complex projective surfaces on which Zariski decomposition can be combined with vanishing theorems to yield cohomology formulae for all line bundles. The obtained formulae express cohomologies in terms…

We analyze the structure of the algebraic manifolds $Y$ of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$, by showing the deformation invariant of some open surfaces. Secondly, we show…

This thesis contributes with a number of topics to the subject of string compactifications. In the first half of the work, I discuss the Hodge plot of Calabi-Yau threefolds realised as hypersurfaces in toric varieties. The intricate…

Motivated by the relation between (twisted) K3 surfaces and special cubic fourfolds, we construct moduli spaces of polarized twisted K3 surfaces of any fixed degree and order. We do this by mimicking the construction of the moduli space of…

Let $X$ be a compact Calabi-Yau 3-fold, and write $\mathcal M,\bar{\mathcal M}$ for the moduli stacks of objects in coh$(X),D^b$coh$(X)$. There are natural line bundles $K_{\mathcal M}\to\mathcal M$, $K_{\bar{\mathcal M}}\to\bar{\mathcal…

Fix a polarised Calabi-Yau threefold $(X,H)$. We reduce a version of the Bayer-Macr\`i-Toda conjecture for $(X,H)$, which ensures the existence of Bridgeland stability conditions on $X$, to verifying a Brill-Noether-type inequality for…

For an abelian or a projective K3 surface $X$ over an algebraically closed field $k$, consider the moduli space $\splcpx_{X/k}\uet$ of the objects $E$ in $D^b(\mathrm{Coh}(X))$ satisfying $\Ext^{-1}_X(E,E)=0$ and $\Hom(E,E)\cong k$. Then we…

We study the geometry of the triplectic quantization of gauge theories. We show that underlying the triplectic geometry is a Kaehler manifold N endowed with a pair of transversal polarizations. The antibrackets can be brought to the…

We establish a structural understanding of the involutive Heegaard Floer homology for all linear combinations of almost-rational (AR) plumbed three-manifolds. We use this to show that the Neumann-Siebenmann invariant is a homology cobordism…

On promoting the type IIA side of the N=1 Heterotic/type IIA dual pairs of [1] to M-theory on a `barely G_2 Manifold' of [2], by spectrum-matching we show a possible triality between Heterotic on a self-mirror Calabi-Yau, M-theory on the…

We construct examples of primitive contractions of Calabi--Yau threefolds with exceptional locus being $ \mathbb{P}^1 \times \mathbb{P}^1$, $\mathbb{P}^2$, and smooth del Pezzo surfaces of degrees $\leq 5$. We describe the images of these…

For a smooth projective curve $X$ over $\mathbb C_p$ and any reductive group $G$, we show that the moduli stack of $G$-Higgs bundles on $X$ is a twist of the moduli stack of v-topological $G$-bundles on $X_v$ in a canonical way. We explain…

We study topological strings on non-commutative resolutions of singular Calabi-Yau threefolds that are double covers of $\mathbb{P}^3$, ramified over determinantal octic surfaces. Using conifold transitions to complete intersections in…

We construct quasi-projective moduli spaces of $K$-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily--Borel compactification and investigate a relation between one-dimensional boundary…

A Finsler metric, whose holomorphic curvature is bounded from above by a negative constant, is constructed on the moduli stack of canonically polarized manifolds including singularities. Demailly's version of Ahlfors' lemma yields…

We describe the proof that the period map from the Torelli space of Calabi-Yau manifolds to the classifying space of polarized Hodge structures is an embedding. The proof is based on the constructions of holomorphic affine structure on the…

We prove that the moduli space of Calabi-Yau 3-folds coming from eight planes of $P^3$ in general positions is not modular. In fact we show the stronger statement that the Zariski closure of the monodromy group is actually the whole…

In a recent preprint, Chi Li proved that aymptotically conical complex manifolds with regular tangent cone at infinity admit holomorphic compactifications (his result easily extends to the quasiregular case). In this short note, we show…

Let $X$ be a smooth complex manifold. Assume that $Y\subset X$ is a K\"{a}hler submanifold such that $X\setminus Y$ is biholomorphic to $\mathbb{C}^n$. We prove that $(X, Y)$ is biholomorphic to the standard example $(\mathbb{P}^n,…

We prove two results regarding Hodge structures appearing in the cohomology of complex tori. First, we prove that if a polarizable Hodge structure appears in the cohomology of a complex torus $T$, it appears in the cohomology of an abelian…